An Introduction to When and How to Conduct a Meta-Analysis
Mike W.-L. Cheung and Ranjith Vijayakumar
25 Feb 2016
Introduction
- This page is a supporting document to Cheung and Vijayakumar (2016). It provides a quick introduction on how to conduct meta-analysis in various software packages. Since different default estimators are used in different packages, the results may be slightly different.
- Cheung, M. W.-L., & Vijayakumar, R. (2016). An introduction to when and how to conduct a meta-analysis. Manuscript in preparation.
Descriptive statistics of the data
- The sample data sets are available as CSV and plain text formats.
- Effect size (
y): standardized mean difference between schizophrenic and control groups. - Sampling variance (
v): Sampling variance ofy. - Moderator (
x): Mean age of the participants.
- IV (x): Mean age of participants
R
- There are several packages to conduct meta-analysis in R. We are going to illustrate the procedures with the metaSEM package.
Fixed-effects model
## You need to install the metaSEM package before using it.
## You need to install it only once.
## install.packages("metaSEM")
## Load the library
library(metaSEM)
## Read the data file
my.df <- read.csv("data.csv")
## Display the first few studies
head(my.df)## y v x
## 1 -1.8586 0.0743 64.92
## 2 -0.7913 0.0545 50.71
## 3 -0.6882 0.0375 39.71
## 4 -0.5261 0.0360 42.21
## 5 -0.4075 0.0412 34.43
## 6 -1.3356 0.0404 54.05## Fixed-effects model by restricting the random effects to 0
summary(meta(y=y, v=v, data=my.df, RE.constraints = 0))##
## Call:
## meta(y = y, v = v, data = my.df, RE.constraints = 0)
##
## 95% confidence intervals: z statistic approximation
## Coefficients:
## Estimate Std.Error lbound ubound z value Pr(>|z|)
## Intercept1 -0.722451 0.028878 -0.779052 -0.665851 -25.017 < 2.2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Q statistic on the homogeneity of effect sizes: 85.71155
## Degrees of freedom of the Q statistic: 49
## P value of the Q statistic: 0.0009179457
##
## Heterogeneity indices (based on the estimated Tau2):
## Estimate
## Intercept1: I2 (Q statistic) 0
##
## Number of studies (or clusters): 50
## Number of observed statistics: 50
## Number of estimated parameters: 1
## Degrees of freedom: 49
## -2 log likelihood: 19.40719
## OpenMx status1: 0 ("0" or "1": The optimization is considered fine.
## Other values may indicate problems.)Random-effects model
summary(meta(y=y, v=v, data=my.df))##
## Call:
## meta(y = y, v = v, data = my.df)
##
## 95% confidence intervals: z statistic approximation
## Coefficients:
## Estimate Std.Error lbound ubound z value Pr(>|z|)
## Intercept1 -0.7286574 0.0373260 -0.8018151 -0.6554997 -19.5214 < 2e-16
## Tau2_1_1 0.0272966 0.0148155 -0.0017412 0.0563344 1.8424 0.06541
##
## Intercept1 ***
## Tau2_1_1 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Q statistic on the homogeneity of effect sizes: 85.71155
## Degrees of freedom of the Q statistic: 49
## P value of the Q statistic: 0.0009179457
##
## Heterogeneity indices (based on the estimated Tau2):
## Estimate
## Intercept1: I2 (Q statistic) 0.3955
##
## Number of studies (or clusters): 50
## Number of observed statistics: 50
## Number of estimated parameters: 2
## Degrees of freedom: 48
## -2 log likelihood: 13.01445
## OpenMx status1: 0 ("0" or "1": The optimization is considered fine.
## Other values may indicate problems.)Mixed-effects model
summary(meta(y=y, v=v, x=x, data=my.df))##
## Call:
## meta(y = y, v = v, x = x, data = my.df)
##
## 95% confidence intervals: z statistic approximation
## Coefficients:
## Estimate Std.Error lbound ubound z value Pr(>|z|)
## Intercept1 -0.4439440 0.1214427 -0.6819674 -0.2059207 -3.6556 0.0002566
## Slope1_1 -0.0071136 0.0029137 -0.0128244 -0.0014029 -2.4414 0.0146292
## Tau2_1_1 0.0214313 0.0132301 -0.0044992 0.0473618 1.6199 0.1052553
##
## Intercept1 ***
## Slope1_1 *
## Tau2_1_1
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Q statistic on the homogeneity of effect sizes: 85.71155
## Degrees of freedom of the Q statistic: 49
## P value of the Q statistic: 0.0009179457
##
## Explained variances (R2):
## y1
## Tau2 (no predictor) 0.0273
## Tau2 (with predictors) 0.0214
## R2 0.2149
##
## Number of studies (or clusters): 50
## Number of observed statistics: 50
## Number of estimated parameters: 3
## Degrees of freedom: 47
## -2 log likelihood: 7.187943
## OpenMx status1: 0 ("0" or "1": The optimization is considered fine.
## Other values may indicate problems.)Mplus
- Mike Cheung provides sample Mplus code on how to conduct meta-analysis in Mplus.
- The sample data set for Mplus is available here.
Fixed-effects model
TITLE: Fixed-effects model
DATA: FILE IS data.dat;
VARIABLE: NAMES y v x;
USEVARIABLES ARE y w2; ! Use both y and w2 in the analysis
DEFINE: w2 = SQRT(v**(-1)); ! Weight for transformation
y = w2*y; ! Transformed effect size
MODEL:
[y@0.0]; ! Intercept fixed at 0
y@1.0; ! Error variance fixed at 1
y ON w2; ! Common effect estimate beta_F
OUTPUT: SAMPSTAT;
CINTERVAL(symmetric); ! Wald CIMplus VERSION 7.4
MUTHEN & MUTHEN
02/20/2016 2:05 PM
INPUT INSTRUCTIONS
TITLE: Fixed-effects model
DATA: FILE IS data.dat;
VARIABLE: NAMES y v x;
USEVARIABLES ARE y w2; ! Use both y and w2 in the analysis
DEFINE: w2 = SQRT(v**(-1)); ! Weight for transformation
y = w2*y; ! Transformed effect size
MODEL:
[y@0.0]; ! Intercept fixed at 0
y@1.0; ! Error variance fixed at 1
y ON w2; ! Common effect estimate beta_F
OUTPUT: SAMPSTAT;
CINTERVAL(symmetric); ! Wald CI
INPUT READING TERMINATED NORMALLY
Fixed-effects model
SUMMARY OF ANALYSIS
Number of groups 1
Number of observations 50
Number of dependent variables 1
Number of independent variables 1
Number of continuous latent variables 0
Observed dependent variables
Continuous
Y
Observed independent variables
W2
Estimator ML
Information matrix OBSERVED
Maximum number of iterations 1000
Convergence criterion 0.500D-04
Maximum number of steepest descent iterations 20
Input data file(s)
data.dat
Input data format FREE
SAMPLE STATISTICS
SAMPLE STATISTICS
Means
Y W2
________ ________
1 -3.568 4.881
Covariances
Y W2
________ ________
Y 1.497
W2 0.092 0.157
Correlations
Y W2
________ ________
Y 1.000
W2 0.190 1.000
UNIVARIATE SAMPLE STATISTICS
UNIVARIATE HIGHER-ORDER MOMENT DESCRIPTIVE STATISTICS
Variable/ Mean/ Skewness/ Minimum/ % with Percentiles
Sample Size Variance Kurtosis Maximum Min/Max 20%/60% 40%/80% Median
Y -3.568 -0.461 -6.819 2.00% -4.377 -3.767 -3.399
50.000 1.497 0.475 -1.134 2.00% -3.282 -2.741
W2 4.881 -0.291 3.669 2.00% 4.623 4.784 4.891
50.000 0.157 0.785 5.882 2.00% 4.933 5.206
THE MODEL ESTIMATION TERMINATED NORMALLY
MODEL FIT INFORMATION
Number of Free Parameters 1
Loglikelihood
H0 Value -88.803
H1 Value -80.122
Information Criteria
Akaike (AIC) 179.605
Bayesian (BIC) 181.517
Sample-Size Adjusted BIC 178.379
(n* = (n + 2) / 24)
Chi-Square Test of Model Fit
Value 17.362
Degrees of Freedom 2
P-Value 0.0002
RMSEA (Root Mean Square Error Of Approximation)
Estimate 0.392
90 Percent C.I. 0.236 0.571
Probability RMSEA <= .05 0.000
CFI/TLI
CFI 0.000
TLI -8.154
Chi-Square Test of Model Fit for the Baseline Model
Value 1.839
Degrees of Freedom 1
P-Value 0.1751
SRMR (Standardized Root Mean Square Residual)
Value 0.322
MODEL RESULTS
Two-Tailed
Estimate S.E. Est./S.E. P-Value
Y ON
W2 -0.722 0.029 -25.017 0.000
Intercepts
Y 0.000 0.000 999.000 999.000
Residual Variances
Y 1.000 0.000 999.000 999.000
QUALITY OF NUMERICAL RESULTS
Condition Number for the Information Matrix 0.100E+01
(ratio of smallest to largest eigenvalue)
CONFIDENCE INTERVALS OF MODEL RESULTS
Lower .5% Lower 2.5% Lower 5% Estimate Upper 5% Upper 2.5% Upper .5%
Y ON
W2 -0.797 -0.779 -0.770 -0.722 -0.675 -0.666 -0.648
Intercepts
Y 0.000 0.000 0.000 0.000 0.000 0.000 0.000
Residual Variances
Y 1.000 1.000 1.000 1.000 1.000 1.000 1.000
DIAGRAM INFORMATION
Use View Diagram under the Diagram menu in the Mplus Editor to view the diagram.
If running Mplus from the Mplus Diagrammer, the diagram opens automatically.
Diagram output
d:\dropbox\aaa\nerv paper\illustrations\mplus1.dgm
Beginning Time: 14:05:23
Ending Time: 14:05:23
Elapsed Time: 00:00:00
MUTHEN & MUTHEN
3463 Stoner Ave.
Los Angeles, CA 90066
Tel: (310) 391-9971
Fax: (310) 391-8971
Web: www.StatModel.com
Support: Support@StatModel.com
Copyright (c) 1998-2015 Muthen & MuthenRandom-effects model
TITLE: Random-effects model
DATA: FILE IS data.dat;
VARIABLE: NAMES y v x;
USEVARIABLES ARE y w2; ! Use both y and w2 in the analysis
DEFINE: w2 = SQRT(v**(-1)); ! Weight for transformation
y = w2*y; ! Transformed effect size
ANALYSIS: TYPE=RANDOM; ! Use random slopes analysis
ESTIMATOR=ML; ! Use ML estimation
MODEL:
[y@0.0]; ! Intercept fixed at 0
y@1.0; ! Error variance fixed at 1
f | y ON w2; ! f: Study specific random effects
f*; ! var(f): tau^2
[f*]; ! mean(f): Average effect size beta_R
OUTPUT: SAMPSTAT;
CINTERVAL(symmetric); ! Wald CIMplus VERSION 7.4
MUTHEN & MUTHEN
02/20/2016 2:09 PM
INPUT INSTRUCTIONS
TITLE: Random-effects model
DATA: FILE IS data.dat;
VARIABLE: NAMES y v x;
USEVARIABLES ARE y w2; ! Use both y and w2 in the analysis
DEFINE: w2 = SQRT(v**(-1)); ! Weight for transformation
y = w2*y; ! Transformed effect size
ANALYSIS: TYPE=RANDOM; ! Use random slopes analysis
ESTIMATOR=ML; ! Use ML estimation
MODEL:
[y@0.0]; ! Intercept fixed at 0
y@1.0; ! Error variance fixed at 1
f | y ON w2; ! f: Study specific random effects
f*; ! var(f): tau^2
[f*]; ! mean(f): Average effect size beta_R
OUTPUT: SAMPSTAT;
CINTERVAL(symmetric); ! Wald CI
INPUT READING TERMINATED NORMALLY
Random-effects model
SUMMARY OF ANALYSIS
Number of groups 1
Number of observations 50
Number of dependent variables 1
Number of independent variables 1
Number of continuous latent variables 1
Observed dependent variables
Continuous
Y
Observed independent variables
W2
Continuous latent variables
F
Estimator ML
Information matrix OBSERVED
Maximum number of iterations 100
Convergence criterion 0.100D-05
Maximum number of EM iterations 500
Convergence criteria for the EM algorithm
Loglikelihood change 0.100D-02
Relative loglikelihood change 0.100D-05
Derivative 0.100D-03
Minimum variance 0.100D-03
Maximum number of steepest descent iterations 20
Optimization algorithm EMA
Input data file(s)
data.dat
Input data format FREE
SAMPLE STATISTICS
ESTIMATED SAMPLE STATISTICS
Means
Y W2
________ ________
1 -3.568 4.881
Covariances
Y W2
________ ________
Y 1.497
W2 0.092 0.157
Correlations
Y W2
________ ________
Y 1.000
W2 0.190 1.000
UNIVARIATE SAMPLE STATISTICS
UNIVARIATE HIGHER-ORDER MOMENT DESCRIPTIVE STATISTICS
Variable/ Mean/ Skewness/ Minimum/ % with Percentiles
Sample Size Variance Kurtosis Maximum Min/Max 20%/60% 40%/80% Median
Y -3.568 -0.461 -6.819 2.00% -4.377 -3.767 -3.399
50.000 1.497 0.475 -1.134 2.00% -3.282 -2.741
W2 4.881 -0.291 3.669 2.00% 4.623 4.784 4.891
50.000 0.157 0.785 5.882 2.00% 4.933 5.206
THE MODEL ESTIMATION TERMINATED NORMALLY
MODEL FIT INFORMATION
Number of Free Parameters 2
Loglikelihood
H0 Value -85.606
Information Criteria
Akaike (AIC) 175.213
Bayesian (BIC) 179.037
Sample-Size Adjusted BIC 172.759
(n* = (n + 2) / 24)
MODEL RESULTS
Two-Tailed
Estimate S.E. Est./S.E. P-Value
Means
F -0.729 0.037 -19.522 0.000
Intercepts
Y 0.000 0.000 999.000 999.000
Variances
F 0.027 0.015 1.842 0.065
Residual Variances
Y 1.000 0.000 999.000 999.000
QUALITY OF NUMERICAL RESULTS
Condition Number for the Information Matrix 0.858E-01
(ratio of smallest to largest eigenvalue)
CONFIDENCE INTERVALS OF MODEL RESULTS
Lower .5% Lower 2.5% Lower 5% Estimate Upper 5% Upper 2.5% Upper .5%
Means
F -0.825 -0.802 -0.790 -0.729 -0.667 -0.656 -0.633
Intercepts
Y 0.000 0.000 0.000 0.000 0.000 0.000 0.000
Variances
F -0.011 -0.002 0.003 0.027 0.052 0.056 0.065
Residual Variances
Y 1.000 1.000 1.000 1.000 1.000 1.000 1.000
DIAGRAM INFORMATION
Use View Diagram under the Diagram menu in the Mplus Editor to view the diagram.
If running Mplus from the Mplus Diagrammer, the diagram opens automatically.
Diagram output
d:\dropbox\aaa\nerv paper\illustrations\mplus2.dgm
Beginning Time: 14:09:42
Ending Time: 14:09:42
Elapsed Time: 00:00:00
MUTHEN & MUTHEN
3463 Stoner Ave.
Los Angeles, CA 90066
Tel: (310) 391-9971
Fax: (310) 391-8971
Web: www.StatModel.com
Support: Support@StatModel.com
Copyright (c) 1998-2015 Muthen & MuthenMixed-effects model
TITLE: Random-effects model
DATA: FILE IS data.dat;
VARIABLE: NAMES y v x;
USEVARIABLES ARE y w2; ! Use both y and w2 in the analysis
DEFINE: w2 = SQRT(v**(-1)); ! Weight for transformation
y = w2*y; ! Transformed effect size
ANALYSIS: TYPE=RANDOM; ! Use random slopes analysis
ESTIMATOR=ML; ! Use ML estimation
MODEL:
[y@0.0]; ! Intercept fixed at 0
y@1.0; ! Error variance fixed at 1
f | y ON w2; ! f: Study specific random effects
f*; ! var(f): tau^2
[f*]; ! mean(f): Average effect size beta_R
OUTPUT: SAMPSTAT;
CINTERVAL(symmetric); ! Wald CIMplus VERSION 7.4
MUTHEN & MUTHEN
02/20/2016 2:09 PM
INPUT INSTRUCTIONS
TITLE: Random-effects model
DATA: FILE IS data.dat;
VARIABLE: NAMES y v x;
USEVARIABLES ARE y w2; ! Use both y and w2 in the analysis
DEFINE: w2 = SQRT(v**(-1)); ! Weight for transformation
y = w2*y; ! Transformed effect size
ANALYSIS: TYPE=RANDOM; ! Use random slopes analysis
ESTIMATOR=ML; ! Use ML estimation
MODEL:
[y@0.0]; ! Intercept fixed at 0
y@1.0; ! Error variance fixed at 1
f | y ON w2; ! f: Study specific random effects
f*; ! var(f): tau^2
[f*]; ! mean(f): Average effect size beta_R
OUTPUT: SAMPSTAT;
CINTERVAL(symmetric); ! Wald CI
INPUT READING TERMINATED NORMALLY
Random-effects model
SUMMARY OF ANALYSIS
Number of groups 1
Number of observations 50
Number of dependent variables 1
Number of independent variables 1
Number of continuous latent variables 1
Observed dependent variables
Continuous
Y
Observed independent variables
W2
Continuous latent variables
F
Estimator ML
Information matrix OBSERVED
Maximum number of iterations 100
Convergence criterion 0.100D-05
Maximum number of EM iterations 500
Convergence criteria for the EM algorithm
Loglikelihood change 0.100D-02
Relative loglikelihood change 0.100D-05
Derivative 0.100D-03
Minimum variance 0.100D-03
Maximum number of steepest descent iterations 20
Optimization algorithm EMA
Input data file(s)
data.dat
Input data format FREE
SAMPLE STATISTICS
ESTIMATED SAMPLE STATISTICS
Means
Y W2
________ ________
1 -3.568 4.881
Covariances
Y W2
________ ________
Y 1.497
W2 0.092 0.157
Correlations
Y W2
________ ________
Y 1.000
W2 0.190 1.000
UNIVARIATE SAMPLE STATISTICS
UNIVARIATE HIGHER-ORDER MOMENT DESCRIPTIVE STATISTICS
Variable/ Mean/ Skewness/ Minimum/ % with Percentiles
Sample Size Variance Kurtosis Maximum Min/Max 20%/60% 40%/80% Median
Y -3.568 -0.461 -6.819 2.00% -4.377 -3.767 -3.399
50.000 1.497 0.475 -1.134 2.00% -3.282 -2.741
W2 4.881 -0.291 3.669 2.00% 4.623 4.784 4.891
50.000 0.157 0.785 5.882 2.00% 4.933 5.206
THE MODEL ESTIMATION TERMINATED NORMALLY
MODEL FIT INFORMATION
Number of Free Parameters 2
Loglikelihood
H0 Value -85.606
Information Criteria
Akaike (AIC) 175.213
Bayesian (BIC) 179.037
Sample-Size Adjusted BIC 172.759
(n* = (n + 2) / 24)
MODEL RESULTS
Two-Tailed
Estimate S.E. Est./S.E. P-Value
Means
F -0.729 0.037 -19.522 0.000
Intercepts
Y 0.000 0.000 999.000 999.000
Variances
F 0.027 0.015 1.842 0.065
Residual Variances
Y 1.000 0.000 999.000 999.000
QUALITY OF NUMERICAL RESULTS
Condition Number for the Information Matrix 0.858E-01
(ratio of smallest to largest eigenvalue)
CONFIDENCE INTERVALS OF MODEL RESULTS
Lower .5% Lower 2.5% Lower 5% Estimate Upper 5% Upper 2.5% Upper .5%
Means
F -0.825 -0.802 -0.790 -0.729 -0.667 -0.656 -0.633
Intercepts
Y 0.000 0.000 0.000 0.000 0.000 0.000 0.000
Variances
F -0.011 -0.002 0.003 0.027 0.052 0.056 0.065
Residual Variances
Y 1.000 1.000 1.000 1.000 1.000 1.000 1.000
DIAGRAM INFORMATION
Use View Diagram under the Diagram menu in the Mplus Editor to view the diagram.
If running Mplus from the Mplus Diagrammer, the diagram opens automatically.
Diagram output
d:\dropbox\aaa\nerv paper\illustrations\mplus2.dgm
Beginning Time: 14:09:42
Ending Time: 14:09:42
Elapsed Time: 00:00:00
MUTHEN & MUTHEN
3463 Stoner Ave.
Los Angeles, CA 90066
Tel: (310) 391-9971
Fax: (310) 391-8971
Web: www.StatModel.com
Support: Support@StatModel.com
Copyright (c) 1998-2015 Muthen & MuthenStata
- There are several user-contributed packages in meta-analysis in Stata.
- The followings files are used in this illustration: data, do-file, and log file.
Fixed- and random-effects models
* Read the data file "data.csv".
. import delim using data.csv(3 vars, 50 obs)* Generate se (standard error) from v (sampling variance).
. generate se=sqrt(v)
* Display the content.
. describeContains data
obs: 50
vars: 4
size: 800
----------------------------------------------------------------------------------------
storage display value
variable name type format label variable label
----------------------------------------------------------------------------------------
y float %9.0g
v float %9.0g
x float %9.0g
se float %9.0g
----------------------------------------------------------------------------------------* Run a meta-analysis on y with se as the standard error.
. meta y seMeta-analysis
| Pooled 95% CI Asymptotic No. of
Method | Est Lower Upper z_value p_value studies
-------+----------------------------------------------------
Fixed | -0.722 -0.779 -0.666 -25.017 0.000 50
Random | -0.729 -0.804 -0.654 -19.030 0.000
Test for heterogeneity: Q= 85.712 on 49 degrees of freedom (p= 0.001)
Moment-based estimate of between studies variance = 0.031Mixed-effects model
* Run a mixed-effects meta-analysis on y with x as the predictor and se as the standard error.
. metareg y x, wsse(se)Meta-regression Number of obs = 50
REML estimate of between-study variance tau2 = .02432
% residual variation due to heterogeneity I-squared_res = 38.44%
Proportion of between-study variance explained Adj R-squared = 15.93%
With Knapp-Hartung modification
------------------------------------------------------------------------------
y | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
x | -.0071645 .0030496 -2.35 0.023 -.0132961 -.0010329
_cons | -.4422977 .1273233 -3.47 0.001 -.6982984 -.1862969
------------------------------------------------------------------------------SPSS
- To use SPSS to conduct meta-analysis, we use the macros provided by David Wilson.
- There are three macros at the site for meta-analysis. We use
MEANESmacro to conduct the basic meta-analysis, and theMETAREGmacro for study-level moderator analysis, using meta-regression.METAFis used to conduct moderator analysis with a categorical moderator. - In our dataset, our study effect size is labeled
y, its sampling variance asv, and moderator Mean Age of the participants asx. - Because the macros require us to input study weights, we first compute a new variable labelled
w, which indicates the weights allotted to each study. We use the formula: w = 1/v to compute this variable. - Once we have the variables
y,x, andwin SPSS, we are ready to use the macros.
Fixed- and random- effects models
SPSS syntax:
INCLUDE 'C:\MEANES.SPS'. MEANES ES = y /W = w.Output:
Run MATRIX procedure:
Version 2005.05.23
***** Meta-Analytic Results *****
------- Distribution Description ---------------------------------
N Min ES Max ES Wghtd SD
50.000 -1.859 -.234 .267
------- Fixed & Random Effects Model -----------------------------
Mean ES -95%CI +95%CI SE Z P
Fixed -.7225 -.7791 -.6659 .0289 -25.0171 .0000
Random -.7292 -.8043 -.6541 .0383 -19.0298 .0000
------- Random Effects Variance Component ------------------------
v = .031257
------- Homogeneity Analysis -------------------------------------
Q df p
85.7115 49.0000 .0009
Random effects v estimated via noniterative method of moments.
------ END MATRIX -----
Mixed-effects model
- There are three estimation methods that can be used: method-of-moments(M-M) which is non-iterative; maximum likelihood (ML); and restricted maximum likelihood (REML).
- These different methods can be called by specifying one of the following commands in SPSS syntax:
METAREG ES = y /W = w /IVS = x /MODEL = MM.METAREG ES = y /W = w /IVS = x /MODEL = ML.METAREG ES = y /W = w /IVS = x /MODEL = REML.
We will use ML to estimate the meta-regression mixed model by the following command:
INCLUDE 'C:\METAREG.SPS' . METAREG ES = y /W = w /IVS = x /MODEL = ML.Output:
Run MATRIX procedure:
Version 2005.05.23
***** Inverse Variance Weighted Regression *****
***** Random Intercept, Fixed Slopes Model *****
------- Descriptives -------
Mean ES R-Square k
-.7277 .1020 50.0000
------- Homogeneity Analysis -------
Q df p
Model 6.0021 1.0000 .0143
Residual 52.8294 48.0000 .2929
Total 58.8314 49.0000 .1587
------- Regression Coefficients -------
B SE -95% CI +95% CI Z P Beta
Constant -.4439 .1212 -.6815 -.2064 -3.6632 .0002 .0000
x -.0071 .0029 -.0128 -.0014 -2.4499 .0143 -.3194
------- Maximum Likelihood Random Effects Variance Component -------
v = .02143
se(v) = .01263
------ END MATRIX -----Comprehensive Meta-Analsyis (CMA)
- CMA provides a graphical user interface to conduct meta-analysis. The program has a regular spread-sheet like window. This makes it easier to copy data from spreadsheets onto this program.
Data Entry
- To initiate a meta-analysis, you can first include the study names into a column specially designated for it, by using the drop-down options as shown below:
- This creates a column for the study names. It is customary to have the first author and year in the study detail. For our purposes, i just labeled studies by numbers (see screenshot below).
- After this, you can input effect sizes. CMA can accept many varieties of effect size parameters, so you need to specify it. We will specify Hedges’s d as our effect size.
The following window pops up. Click ‘Next’.
- You can now select the kind of studies you are analyzing. Here, since we are comparing studies that compare schizophrenics with a control group, we choose the first option, and then click ‘Next’. See below.
- A new window appears, from which you can choose the desired effect size metric. Our metric is Hedge’s d, so we click on ‘continuous means’, which gives us the option, ‘computed effect sizes’, which then gives us a lot of effect sizes, from which we choose Hedges’s d with variance. See below.
- The main window now appears with columns where we can input the effect sizes of individual studies. We can also optionally label our groups as well. We label them as ‘Schizophrenia’ and ‘Control’, as below.
- We can now transfer our effect sizes from spreadsheets onto this window. The window after transferring our effect size and variance column is as below.
- As shown above, you have to specify the direction (sign) of the effect size. For each study we specify it to be ‘auto’; it then takes the sign of our input Hedges’s d.
- You can also see that the yellow-highlighted columns are calculated outputs by CMA. In our case, since we have given the effect sizes already, this seems redundant. But you can instead opt to input just the means and SDs of the two groups; CMA will calculate the effect size and display it in the yellow column.
- We are also interested in looking at the effect of age as a moderator. So we will include the age in the dataset as follows:
- We then get a new window where we can specify the variable name and data type. We specify ‘Age’ as the moderator.
- We get the final dataset with both the effect size and moderator columns, as shown below.
- Now we can conduct our analysis. We will do both a meta-analysis using fixed effects model as well as random-effects model. We will then do a meta-regression, with age as the moderator.
Fixed- and mixed-effects models
- Click the ‘run analyses’ icon, as shown below.
- The default analysis is fixed effects model, so you get the display below.
- Clicking on “Both models” icon (see figure below) will show both fixed-effects and random-effects models.
- Clicking on ‘View’, as shown below, brings up the ‘statistics’ option.
- That brings up the window below. It gives both the fixed-effects as well as random-effects results.
Mixed-effects model
- Click on ‘analyses’ and then ‘meta regression 2’ icons as shown below:
- We get the screen below. This screen shows all variables we had included in the main window spreadsheet under the label ‘moderator’. The window shows that currently, we do not have any moderators in our meta-analysis model. We specify that we should include ‘age’ in our model, by clicking on ‘Age’, and then ‘Add to main screen.’
- Then, we specify the type of estimation method we want to employ (‘Maximum Likelihood’), make sure the “Random” icon at the lower left is highlighted to specify a mixed model, and then click ‘Run Regression’. See below.
- We get the results below:
- We can get more details by clicking on the tabs at the top of the window. For instance, clicking on “Scatterplot” gives you the following figure:
